business economics [1.5ex] elasticity [-0.7ex] and its ...arc-elasticity fordiscretechanges: e.g.:...

Business Economics

Elasticityand its applications

Thomas & Maurice, Chapter 6

Herbert Stocker

[email protected]

Institute of International StudiesUniversity of Ramkhamhaeng

&Department of EconomicsUniversity of Innsbruck

Elasticities

Question: How ‘strongly’ reacts one variable inresponse of a change in another variable?

→ slope of a curve.

e.g. Demand for wheat in the USA:

Qd = 3550− 266P

where Qd is measured in million ‘bushels’ per yearand P is measured in US$.

Slope: dQd

dP= −266

i.e. if the price increases by one dollar the quantitydemanded decreases by 266 mio ‘bushels’!???

Elasticities

Problem: For the interpretation of the slope wehave to know the dimensions in which the unitsare measured!Alfred Marshall (1842-1924):proposed the use of relative changes!

Price-elasticity =Percentage change in quantity

Percentage change in price

=%∆Qd

%∆P

e.g. what is the percentage decrease in thequantity demanded of wheat if the price of wheatraises by one percent?

Elasticities

Elasticity: Percentage change in the dependentvariable resulting from a one percent increase inthe independent variable.

Elasticities are a very general concept to expressthe ‘strength of reaction’ of one variable inresponse to the change of another variable.

Main advantage of elsticities: free of

dimensions!

Two concepts:Arc-Elasticity: discrete change between two observedpoints.Point-Elasticity: infinitesimal change (function mustbe known).

Arc-Elasticity

When only two price-quantity pairs are known:(for discrete changes)

EB =%-Change of Y

%-Change of X

=

Y2−Y1

Y1

X2−X1

X1

≡∆YY1

∆XX1

≡∆Y∆XY1

X1

=∆Y

∆X

X1

Y1

Y

X

b before

X1

Y1

bc after

X2

Y2

∆X

∆Y

Arc-elasticity

For discrete changes:

e.g.: price raises from 2$ to 4$ ⇒ quantity decreasesfrom 20 kg to 15 kg.

P

Qd

bc

b∆P

P2 = 4

P1 = 2

bc

bc

∆Qd

15 = Qd2 Qd1 = 20

∆Qd = Qd1 − Qd2

= 20− 15 = 5

∆P = P1 − P2

= 2− 4 = −2

EBQd ,P

=

∆Qd

Qd1

∆PP1

=520−22

= −1

4

Arc-Elasticity

Problem: the values of the arc-elasticity dependon whether the price increases or decreases!

E 1Qd ,P

=

Qd1−Qd2

Qd1× 100

P1−P2

P1× 100

↔ E 2Qd ,P

=

Qd2−Qd1

Qd2× 100

P2−P1

P2× 100

Solution: Mid-Point Method

EQd ,P =

Qd1−Qd2

(Qd1+Qd2)/2× 100

P1−P2

(P1+P2)/2× 100

Point-Elasticity

For curvilinear demand curves: Slope is calculatedby using the derivative (∆ → d).

Point Elasticity:

EQd ,P =dQd

dPQd

P

=dQd

dP

P

Qd

P

Qd

bc

Qd1

P1b

Point-Elasticity

Example:

What is the price elasticity of demand for the followingdemand function:

Qd = 25− 2.5P

at the price P1 = 2, (⇒ Qd1 = 25− 2.5× 2 = 20):

EQd ,P =dQd

dP

P1

Qd1

= − 2.52

20= − 0.25

Point-Elasticity

Example Qd = 25− 2.5P

at a different price P2 = 4 demand is Qd2 = 15, andthe elasticity is therefore

EQd ,P =dQd

dP

P2

Qd2

= −2.54

15

= −2

3≈ −0.666̇

For linear functions, the value of the elasticity isdifferent at each point!

Point-Elasticity

Problem: For linear functions the elasticity is notconstant along the line!

EQd ,P =dQd

dP

Pi

Qdi

Solution: the elasticity is often calculated at themean value of the variable.

EQd ,P =dQd

dP

P

Qd

mit: P ≡ 1N

∑Ni=1 Pi und Qd ≡ 1

N

∑Ni=1Qdi

Elasticities

An elasticity shows the percentage change in thedependent variable (Y ) when the independent variable(X ) increases by one percent.

Elasticities are positive if the derivative is positive,i.e. if the variables move in the same direction.

Elasticities are negative if the variables move inopposite directions.

An elasticity measures how ‘strongly’ one variablereacts in response of a change in another variable.

Rule of Thumb

Managers can get a rough estimate of price elasticityby asking two questions:

What price P customers you currently pay for theproduct?At what price A would customers stop buying myproduct altogether?The answers to this questions can be used tocalculate a rough estimate of the demandelasticity, since

EQd ,P =P

(P − A)

where P is the price and A is the vertical interceptof the plotted demand curve (the P-axis).

Rule of Thumb

Why?

P

Q

A

bP

P = A− s Q

Q =1

s[A− P]

dQ

dP=

−1

s

E =dQ

dP

P

Q

=−1

s

P1s[A− P]

=P

P − A

Graphical Derivation

Elasticity and Angle

Slope and Angle

Remember:

Hypotenuse

Adjacent

Oppositeleg

α

tanα =Opposite leg

Adjacent

= Slope


Y

X

Abc

α

XA

XA

YA YA

β

Y = f (X )dY

dX= tanα

YA

XA

= tan β

EA =dYdXYA

XA

= −tanα

tan β


Y

X

αβ

Abc

EA =dYdXYAXA

= −tanαtanβ

Y = f ( X )

XA

YA


P

Qd

α

β

Bbc

EA =dQddPQdBPB

= −tanαtan β

Qd = f ( P )

QdB

PB


0

1

2

3

4

0 1 2 3 4

P

Qd

α

bc

β

E1 = − tanαtan β = −

4431

= − 13

bc

γ

E2 = − tanαtan γ = −

4422

= −1

bc

δ

E3 = − tanαtan δ = −

4413

= −3

bc E4 = − tanαtan(0) = −

4404

= − 10 = −∞

bcE0 = − tanα

tan(90) = −4440

= − 1∞

= 0


P

Qd

α

β

A

bc

Positive Intercept:

tanα > tan β

EA =dQd

dPQdA

PA

=tanα

tan β> 1

Question: which value has the elasticity of a linearfunction that goes through the origin?

Functions with constant elasticity

Y = f (X ) = AX b

dY

dX= bAX b−1

Y

X=

AX b

X= AX b−1

EY ,X =dYdXYX

=bAX b−1

AX b−1= b

⇒ Elasticities of power-functions are always constant!

Functions with constant elasticity

Example:

Y = f (X ) = 3X 0.5

dY

dX= 0.5× 3X 0.5−1 = 0.5× 3X−0.5

Y

X=

3X 0.5

X= 3X 0.5−1 = 3X−0.5

EY ,X =dYdXYX

=0.5× 3X−0.5

3X−0.5= 0.5

Elasticities of log-linear Functions

lnY = f (lnX )

EY ,X =d lnY

d lnX=

dYYdXX

=dY

dX

X

Y

Example:

Y = f (X ) = 3X 0.5

lnY = ln 3 + 0.5 lnX

EY ,X =d lnY

d lnX= 0.5


EY ,X =d lnY

d lnX=

dYYdXX

Intuition:

d lnY

dY=

1

Y⇒ d lnY =

dY

Yd lnX

dX=

1

X⇒ d lnX =

dX

X

EY ,X =dYYdXX

=d lnY

d lnX


Example:

Y = AX b

lnY = lnA+ b lnX

EY ,X =d lnY

d lnX= b

Example:

Y = f (X ) = 0.25X−2

lnY = ln 0.25− 2 lnX

EY ,X =d lnY

d lnX= −2

Demand with constant elasticity

P

Qd

Qd = 2P−0.5

Qd = 0.25P−2

Elastic:

Qd = 0.25P−2

lnQd = ln 0.25− 2 lnP

EQd ,P = −2

Inelastic:

Qd = 2P−0.5

lnQd = ln 2− 0.5 lnP

EQd ,P = −0.5

Applications of Elasticities

Theory of Demand

Price Elasticity of Demand

Demand Function:

Qd = Qd(P ,M ,PS ,PC , . . .)

Price Elasticity of Demand:(sometimes called price or demand elasticity)

EQd ,P =dQd

dPQd

P

=dQd

dP

P

Qd

≈%∆Qd

%∆P

The Price Elasticity of Demand shows the

percentage decrease of the quantity demanded if price

ceteris paribus increases by one percent.


Demand is elastic if |EQd ,P | > 1, or

%∆Q > %∆P

(The quantity demanded responds more than proportionally to a a

change in price.

Demand is unit elastic if |EQd ,P | = 1, or

%∆Q = %∆P

Demand is inelastic if 0 < |EQd ,P | < 1, or

%∆Q < %∆P

(The quantity demanded responds less than proportionally to a a

change in price.

Since the price elasticity is usually negative it is common to use the absolutevalue |EQd ,P |.

Linear Demand Function and Elasticity

In which point of a linear demand function theelasticity has the value −1?

elastic

inelastic

bc

bc

bc

EQd ,P = −1

EQd ,P = −∞

EQd ,P = 0

a2b

a2

a

P

Qd

Qd = a − bP

EQd ,P =dQd

dP

P

Qd

=−bP

a − bP= −1

bP = a − bP

P =a

2b

Qd = a − b( a

2b

)

=a

2

Special cases . . .

Perfectly inelastic de-mand: EQd ,P = 0

P

Qd

A change in price has no effect on

the quantity demanded!

Perfectly elastic demand:EQd ,P = −∞

P

Qd

A change in price has an infinitely

large effect on the quantity de-

manded!!

Determinants of price elasticity

Determinants of price elasticity: ceteris paribus

demand tends to be more elastic, . . .

the more and closer substitutes are available.

when the good is rather a luxury than a necessity.

the higher the proportion of income spent on thegood.

the longer the time period under consideration.

demand for durable goods tends to be more elasticthan demand for non-durables (consumers chooseto hold on to the good instead of replacing it).

Price elasticities for cars

Model Price Estimated EQd ,P

Mazda 323 $ 5,039 −6.358Nissan Sentra $ 5,661 −6.528Ford Escort $ 5,663 −6.031Honda Accord $ 9,292 −4.798Ford Taurus $ 9,671 −4.220Nissan Maxima $13,695 −4.845Cadillac Sevifle $24,544 −3.973Lexus LS400 $27,544 −3.085BMW 735i $37,490 −3.515

Source: Table V in S. Berry, Levinsohn, and A. Pakes, “Automobile Prices in Market Equilibrium”, Econometrica 63 (July1995): 841-890. [aus: D. Besanko & D. Braeutigam, Microeconomics (Wiley)]

⇒ probably cheaper cars are perceived more as substitutes than luxury cars.

Price Elasticity

Attention:

Even if the demand for the entire product israther inelastic the elasticity for the individualproducer might be quite large.

For example, the demand for eggs or potatoes israther inelastic, but the elasticity for the eggs orpotatoes of an individual farmer might be close toinfinity!


and Total Revenue

Elasticity & Total Revenue

Elastic Demand:P

Qd

bc

bc

Total revenue decreases

when price increases!

Inelastic Demand:P

Qd

bc

bc

Total revenue increases

when price increases!Total revenue is P × Qd (i.e. the hatched area)


elastic

inelastic

E = −1

E = −∞

E = 0

a2

P

Qd

bc

bc

If demand is ineleas-tic price and totalrevenue move in thesame direction!


elastic

inelastic

E = −1

E = −∞

E = 0

a2

P

Qd

bc

bc

If demand is elasticprice and total rev-enue move in oppositedirection!


With Calculus:

TR = P × Q(P)

d(TR)

dP=

d [P × Q(P)]

dP

= Q + PdQ

dP

= Q

(

1 +dQ

dP

P

Q

)

= Q(1 + EQd ,P)

= Q(1− |EQd ,P |)


d(TR)

dP= Q(1− |EQd ,P |)

d(TR)

dP= 0 for |EQd ,P | = 1 ⇒ Max.!

d(TR)

dP< 0 for |EQd ,P | > 1 ⇒ elastic

d(TR)

dP> 0 for |EQd ,P | < 1 ⇒ inelastic

Other Elasticities

Income Elasticity

Demand function: Qd = Qd(P ,M ,PS ,PC , . . .)

Income Elasticity of Demand:

EQd ,M =Percentage change in quantity demanded

Percentage change in income

=dQd

dMQd

M

=dQd

dM

M

Qd

≈%∆Qd

%∆M

The Income Elasticity of Demand shows the

percentage change in quantity demanded if income

ceteris paribus increases by one precent.

Income Elasticity of Demand

Normal goods (necessities): 0 < EQd ,M < 1:income elasticity is between 0 and 1.

Luxury or superior goods: EQd ,M > 1:if income ceteris paribus increases by one percentthe quantity demanded will increase by more thanone percent! Example: lobster, . . .

Inferior goods: EQd ,M < 0:the quantity demanded decreases if incomeincreases! Example: second-hand clothes, rice, . . .


Income elasticity of demand can be important forfirms:

Demand for luxuries increases more thanproportional with income, markets for luxuriestend to grow more rapidly than markets fornormal and inferior goods.

Firms can try to target marketing campaigns toconsumer groups with higher income elasticity.

Developing countries are often specialized inprimary production with low income elasticities.


Estimates of the Income Elasticity of Demand forSelected Food ProductsProduct Estimated EQd ,M Product Estimated EQd ,M

Cream 1.72 Milk 0.50Peaches 1.43 Butter 0.37Apples 1.32 Potatoes 0.15Fresh peas 1.05 Margarine −0.20Oranges 0.83 Flour −0.36Eggs 0.44

Source: Daniel B. Suits, “Agriculture”, in: The Structure of American Industry, W. Adams and J. Brock, eds.(Englewood, Nj: Prentice Hall), 1995;H. S. Houthhakker and Lester D. Taylor, “Consumer Demand in the United States, 1929-1970” (Cambridge, MA:Harvard University Press), 1966.taken from: D. Besanko & D. Braeutigam, Microeconomics (Wiley)

Cross Price Elasticity

Cross-price elasticity of demand: measureshow demand for Good X varies with changes inthe price of another Good Y .

Substitute goods have positive cross elasticity.Complementary goods have negative cross elasticity.

Defines relevant market in which differentproducts compete.


Demand function: Qd = Qd(P ,M ,PS ,PC , . . .)

Cross Price Elasticity of Demand:

EQd ,PS=

Percentage change in demand for good A

Percentage change in price for good B

=

dQdA

dPB

QdA

PB

=dQdA

dPB

PB

QdA

≈%∆QdA

%∆PB

The Cross Price Elasticity of Demand shows the

percentage change in the demand for a good, if the

price of another good changes by one percent.


Substitutes: (e.g. Cafe and Tea)⇒ cross price elasticity is positiveif tea becomes more expensive the demand for cafe

increases.

EQd ,PS=

dQd

dPS

PS

Qd

> 0

Complementary Goods: (e.g. cafe and sugar)⇒ cross price elasticity is negativeif cafe becomes more expensive the demand for sugar

decreases.

EQd ,PC=

dQd

dPC

PC

Qd

< 0

Price and Cross Price Elasticities

Demand for Price of Beef Price of Pork Price of Chicken

Beef −0.65 0.01 0.20Pork 0.25 −0.45 0.16Chicken 0.12 0.20 −0.65

Source: Daniel B. Suits, “Agriculture”, in: The Structure of American Industry, W. Adams and J. Brock, eds.(Englewood, Nj: Prentice Hall), 1995entnommen aus: D. Besanko & D. Braeutigam, Microeconomics (Wiley)

Price elasticities are on the main diagonale, off the main diagonal are thecross price elasticities.e.g.: −0.65 is the price elasticity of beef,0, 01 is the cross price elasticity of the demand for beef with respect to theprice of pork.

Price and Cross Price Elasticities

Sometimes useful to judge whether markets are‘related’.

Price of Price of Price of Price ofDemand for Sentra Escort LS400 735iSentra −6.528 0.078 0.000 0.000Escort 0.454 −6.031 0.001 0.000LS400 0.000 0.001 −3.085 0.093735i 0.000 0.001 0.032 −3.515

Source: S. Berty Levinsohn, and A. Pakes, “Automobile Prices in Market Equilibrium”, Econometrica 63 (July 1995):841-890.entnommen aus: D. Besanko & D. Braeutigam, Microeconomics (Wiley)

Diagonal elements: the price elasticity of demandOff-diagonal elements: the cross-price elasticity of demand.

Demand for Coca- and Pepsi Cola

Econometrically estimated demand functions:

Qdc = 26.17− 3.98Pc + 2.25Pp + 2.60Ac − 0.62Ap + 0.99M + . . .

Qdp = 17.48− 5.48Pp + 1.40Pc + 2.83Ap − 4.81Ac + 1.92M + . . .

Qdc quantity demanded of Coca-Cola (ten million cases)Qdp quantity demanded of Pepsi (ten million cases)Pc price of Coca-Cola (dollars per ten cases)Pp price of Pepsi (dollars per ten cases)Ac advertising expenditures on behalf of Coca-ColaAp advertising expenditures on behalf of PepsiM disposable income in the United States

All prices expressed in 1986 U.S. dollars!

Elasticities: Coca Cola und Pepsi ColaBy inserting the means (e.g. Pc = 12, 96, Pp = 8, 16; Ac = 5, 89; . . . ) one cancalculate the elasticities in the mean:

Price, Cross-Price, and Income Elasticities of Demandfor Coca-Cola and Pepsi

Elasticity Coca-Cola PepsiPrice elasticity of demand −1.47 −1.55Cross-price elasticity of demand 0.52 0.64Income elasticity of demand 0.58 1.38

Source: Gasmi, F., J.J. Laffont and Q. Vuong (1992): “Econometric Analysis of Collusive Behaviour in the Soft DrinkMarket”, Journal of Economics and Marketing Strategy, Vol. 1entnommen aus: D. Besanko & D. Braeutigam, Microeconomics (Wiley)

Advertising Elasticity of Demand

Advertising elasticity of demand: the percentagechange in quantity demanded of a good relative tothe percentage change in advertising dollars spenton that good.

Marketing studies: e.g. Tellis, 1988; Sethuramanand Tellis, 1991; Hoch, et al, 1995

Advertising elasticities of demand tend to bemuch smaller than price elasticities of demand (bya factor 10-15).

Supply Elasticity

Price elasticity of supply is the percentage change inquantity supplied resulting from a percent change inprice.

ES ,P =dSdPSP

=dS

dP

P

S≥ 0

Supply is elastic when ES ,P > 1

Supply is inelastic when ES ,P < 1

Supply Elasticity

Determinants of Elasticity of Supply:

Time period: Supply is more elastic in the longrun!

Ability of sellers to change the amount of thegood they produce.(Beach-front land is probably inelastic, whilebooks, cars, or manufactured goods are ratherelastic)

Special cases . . .

Perfectly Inelastic Supply:

ES ,P = 0

P

S

An increase in price leaves the

quantity supplied unchanged!

Perfectly Elastic Supply:

ES ,P = ∞

P

Qs

Below the price the quantity sup-plied is zero, above it is infinite!

Price Elasticity in Marketing

Managerial Price Sensitivity Analysis

The price elasticity of demand (in marketingliterature often called price sensitivity) is one ofthe most important variables for managers.

A managerial analysis of price elasticity should bea written document that can be criticized andimproved over time.

It should include some of the following questions:

Reference Price

Substitutes and Reference Price:

Are there close substitutes to the product, and if so,are the buyers (or a segment of buyers) usually awarethereof when making a purchase? Can they compareprices?Can buyers speed up or delay purchases based onexpectations of future prices?

How difficult is it for buyers to compare offers ofdifferent suppliers?

Switching Cost & Expenditure Share

To what extent have buyers already madeinvestments (monetary and/or psychological) thatthey would need to incur again if they switchedsuppliers?

For how long are buyers presumably ‘locked’ bythose expenditures?How significant are buyers expenditures for theproduct?

For end consumers mainly the portion of income isimportant.For business customers also the absolute price mightbe important.

Fairness

Buyers are more sensitive to a product’s pricewhen it is outside the range that they perceive as‘fair’ or ‘reasonable’.

How does the current price compare with pricespeople have paid in the past?

What do buyers expect to pay for similarproducts?

Do customers perceive the product as ‘necessity’or as a discretionary purchase?

Framing Effect

Prospect Theory: (D. Kahneman & A. Tversky)Essential idea: people ‘frame’ purchasing decisionsin their minds as a bundle of gains and losses.

Consumers tend to be more price sensitive whenthey perceive the price as a ‘loss’ rather than aforegone ‘gain’.

Additionally, they are more price sensitive whenthe price is paid separately rather than as part ofa bundle.

Framing Effect

Example: (Prospect Theory)Gas station A sells gasoline for $1.20 and gives a$0.10 per liter discount if the buyer pays with cash.Gas station B sells gasoline for $1.10 and charges a$0.10 surcharge if the buyer pays with credit card.

Most people choose station A.

We’ll have to say a lot more about priceelasticities when we study optimal pricing onimperfect markets.

business economics [1.5ex] elasticity [-0.7ex] and its ...arc-elasticity fordiscretechanges: e.g.:...

Documents